First Law of Thermodynamics
An introduction to the first law of thermodynamics, and discussing the general energy equation and its application to particular cases.
Key FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
Definitions Potential energy (PE) is the energy possessed by a quantity of matter by virtue of its position above some specified datum. Kinetic energy (KE) is the energy possessed by a quantity of matter by virtue of its translational velocity. Internal energy is the sum of the molecular KE (translational and rotational) and the PE of individual molecules. Heat engine is any machine that can convert heat into work, and vice versa. Working substance (WS). The WS is used as the carrier for heat energy. The heat engine carries out the conversion process by a series of changes of state of the WS. The state of the WS is defined by the values of its properties, e.g. pressure, volume, temperature, internal energy, enthalpy. These properties are also sometimes called functions of state. <br/> Key facts The first law of thermodynamics states that energy can neither be created nor destroyed, but only transformed from one form (e.g. work, heat, KE, PE, internal energy) to another. The first law of thermodynamics can be used to develop the general energy equation, which can be written as:
where , , , and , , are the energies, and functions of state respectively characterizing the first state, , , , and , , are the energies, and functions of state respectively characterizing the second state, is the heat supplied to the WS, and the work done by the WS. The general energy equation can also be expressed in terms of enthalpies as:
where is the enthalpy characterizing the first state, while is the enthalpy characterizing the second state. The form of the general energy equation can be adapted to particular cases. For example, for constant volume processes it can be reduced to:
for constant pressure non-flow processes:
for constant pressure flow processes:
for adiabatic non-flow processes:
for adiabatic flow processes:
while for throttling processes it states that there is no change in enthalpy:
The first law of thermodynamics, also sometimes called the law of conservation of energy, states that energy can neither be created nor destroyed, but only transformed from one form to another. The different interchangeable forms of energy include, among others, work, heat, potential energy (PE), kinetic energy (KE), or internal energy.
The General Energy EquationThe first law of thermodynamics can be used to develop the so-called general energy equation for open systems. Imagine a heat engine taking in a working substance (WS) in an initial state characterized by the energies , , and (see Figure 1). The WS passes through the machine having units of heat supplied to it and giving a work output of units. The WS then leaves the machine with energies , , and . The work done to drive one unit mass of air through the system, this time diagramed as in Figure 2, is given by: 3) thus leads to: 4), we can write that: 5) can also be written as: 7) becomes: 7) becomes: 6) becomes:
The Application Of The First Law To Particular CasesAs we previously saw, the first law of thermodynamics can be expressed in terms of the general energy equation (see equations 6 and 7). Also as previously noted, the form of the general energy equation can be adapted to particular applications (see equations 8, 10, and 12). We can further apply the general energy equation to more comprehensive particular cases. 1) Constant Volume Processes In this case, the work output is zero, and the processes are invariably non-flow. Nearly all the terms in the general energy equation (6) are eliminated, leaving: 6) most terms are eliminated, leaving: 19), equation (21) becomes: 7) becomes: 6) becomes: 7) can be written in this case as: 7) becomes in this case:
Example - Horsepower of a gas turbine
Twenty pounds of gas flow per second through a gas turbine. The inlet pressure is (), the specific volume is , and the inlet velocity is . In passing through the engine, the internal energy drops by , while are lost through radiation. If the gas is discharged at and at with a specific volume of , find the horsepower () developed by the turbine.
From the general energy equation we can write that: 2) we have to analyze each individual term. From the hypothesis we know that: 6) we thus obtain that: 3), (5), (8), and (10) in equation (2), we can calculate as: 12), we get the horsepower output of the turbine as: